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Theorem nmvs 18714
Description: Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnlm.v  |-  V  =  ( Base `  W
)
isnlm.n  |-  N  =  ( norm `  W
)
isnlm.s  |-  .x.  =  ( .s `  W )
isnlm.f  |-  F  =  (Scalar `  W )
isnlm.k  |-  K  =  ( Base `  F
)
isnlm.a  |-  A  =  ( norm `  F
)
Assertion
Ref Expression
nmvs  |-  ( ( W  e. NrmMod  /\  X  e.  K  /\  Y  e.  V )  ->  ( N `  ( X  .x.  Y ) )  =  ( ( A `  X )  x.  ( N `  Y )
) )

Proof of Theorem nmvs
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlm.v . . . . 5  |-  V  =  ( Base `  W
)
2 isnlm.n . . . . 5  |-  N  =  ( norm `  W
)
3 isnlm.s . . . . 5  |-  .x.  =  ( .s `  W )
4 isnlm.f . . . . 5  |-  F  =  (Scalar `  W )
5 isnlm.k . . . . 5  |-  K  =  ( Base `  F
)
6 isnlm.a . . . . 5  |-  A  =  ( norm `  F
)
71, 2, 3, 4, 5, 6isnlm 18713 . . . 4  |-  ( W  e. NrmMod 
<->  ( ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing )  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y )
) ) )
87simprbi 452 . . 3  |-  ( W  e. NrmMod  ->  A. x  e.  K  A. y  e.  V  ( N `  ( x 
.x.  y ) )  =  ( ( A `
 x )  x.  ( N `  y
) ) )
9 oveq1 6090 . . . . . 6  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
109fveq2d 5734 . . . . 5  |-  ( x  =  X  ->  ( N `  ( x  .x.  y ) )  =  ( N `  ( X  .x.  y ) ) )
11 fveq2 5730 . . . . . 6  |-  ( x  =  X  ->  ( A `  x )  =  ( A `  X ) )
1211oveq1d 6098 . . . . 5  |-  ( x  =  X  ->  (
( A `  x
)  x.  ( N `
 y ) )  =  ( ( A `
 X )  x.  ( N `  y
) ) )
1310, 12eqeq12d 2452 . . . 4  |-  ( x  =  X  ->  (
( N `  (
x  .x.  y )
)  =  ( ( A `  x )  x.  ( N `  y ) )  <->  ( N `  ( X  .x.  y
) )  =  ( ( A `  X
)  x.  ( N `
 y ) ) ) )
14 oveq2 6091 . . . . . 6  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1514fveq2d 5734 . . . . 5  |-  ( y  =  Y  ->  ( N `  ( X  .x.  y ) )  =  ( N `  ( X  .x.  Y ) ) )
16 fveq2 5730 . . . . . 6  |-  ( y  =  Y  ->  ( N `  y )  =  ( N `  Y ) )
1716oveq2d 6099 . . . . 5  |-  ( y  =  Y  ->  (
( A `  X
)  x.  ( N `
 y ) )  =  ( ( A `
 X )  x.  ( N `  Y
) ) )
1815, 17eqeq12d 2452 . . . 4  |-  ( y  =  Y  ->  (
( N `  ( X  .x.  y ) )  =  ( ( A `
 X )  x.  ( N `  y
) )  <->  ( N `  ( X  .x.  Y
) )  =  ( ( A `  X
)  x.  ( N `
 Y ) ) ) )
1913, 18rspc2v 3060 . . 3  |-  ( ( X  e.  K  /\  Y  e.  V )  ->  ( A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y ) )  -> 
( N `  ( X  .x.  Y ) )  =  ( ( A `
 X )  x.  ( N `  Y
) ) ) )
208, 19syl5com 29 . 2  |-  ( W  e. NrmMod  ->  ( ( X  e.  K  /\  Y  e.  V )  ->  ( N `  ( X  .x.  Y ) )  =  ( ( A `  X )  x.  ( N `  Y )
) ) )
21203impib 1152 1  |-  ( ( W  e. NrmMod  /\  X  e.  K  /\  Y  e.  V )  ->  ( N `  ( X  .x.  Y ) )  =  ( ( A `  X )  x.  ( N `  Y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083    x. cmul 8997   Basecbs 13471  Scalarcsca 13534   .scvsca 13535   LModclmod 15952   normcnm 18626  NrmGrpcngp 18627  NrmRingcnrg 18629  NrmModcnlm 18630
This theorem is referenced by:  nlmdsdi  18719  nlmdsdir  18720  nlmmul0or  18721  lssnlm  18738  nmoleub2lem3  19125  nmoleub3  19129  cphnmvs  19155  nmmulg  24354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-nlm 18636
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